Theorems · Theorem · group theory
Submonoid.mem_closure_iff_exists_finset_subset
∀ {M : Type u_1} [inst : CommMonoid M] {x : M} {s : Set M},
x ∈ Submonoid.closure s ↔ ∃ f t, ↑t ⊆ s ∧ Function.support f ⊆ ↑t ∧ ∏ a ∈ t, a ^ f a = x- Cited by
- 1 results in Mathlib
- Foundations
- Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommMonoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites34
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Finsetstatement and proof · cited by 13,712
- SetLike.coestatement and proof · cited by 8,199
- LE.le.transproof · cited by 3,151
- Submonoidstatement · cited by 3,086
- Finset.prodstatement and proof · cited by 2,356
- CommMonoidstatement and proof · cited by 2,264
- pow_zeroproof · cited by 1,094
- pow_oneproof · cited by 894
- Finset.prod_congrproof · cited by 646
- Function.supportstatement and proof · cited by 610
- Pi.singleproof · cited by 518
Cited by1
Results whose statement or proof uses this declaration.
- Submonoid.mem_closure_finsetproof · cited by 1